We introduce a subfamily of skew Dyck paths called box paths and show that they are in bijection with pairs of ternary trees, confirming an observation stated previously on the On-Line Encyclopedia of Integer Sequences. More generally, we define k-box paths, which are in bijection with (k + 1)-tuples of (k + 2)-ary trees. A bijection is given between k-box paths and a subfamily of k_t-Dyck paths, as well as a bijection with a subfamily of (k,l)-threshold sequences. We also study the refined enumeration of k-box paths by the number of returns and the number of long ascents. Notably, the distribution of long ascents over k-box paths generalizes the Narayana distribution on Dyck paths, and we find that (k − 3)-box paths with exactly two long ascents provide a combinatorial model for the second k-gonal numbers.